Abstract

We extend our theory of on-axis beam scintillations [Waves Random Media 4, 243 (1994)] for the case of propagation on slant turbulent paths, where turbulence is concentrated in a relatively thin layer near the transmitter. Our technique is based on the parabolic equation for optical wave propagation and the Markov approximation for the calculation of statistical moments of beam intensity. This first of two companion papers presents the details of the path integral formulation of the solution for the fourth-order coherence function. We also discuss in detail two analytic techniques that can be used for the treatment of the path integrals.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. J. Baker, “Gaussian beam weak scintillation: Low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006).
    [CrossRef]
  2. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
    [CrossRef]
  3. M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203-1–734203-8 (2009).
  4. V. A. Banakh and I. N. Smalikho, “Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 233–237 (1992).
  5. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4. Wave Propagation through Random Media (Springer, 1988).
  6. M. I. Charnotskii, “Asymptotic analysis of finite beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  7. M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of 5th Symposium on Laser Propagation in the Atmosphere, (USSR Acad. Sci., Siberia, 1979), pp. 74–78.
  8. M. I. Charnotskii, “Beam scintillations for ground-to-space propagation II. Gaussian beam scintillation,” J. Opt. Soc. Am. A 27, 2180–2187 (2010).
    [CrossRef]
  9. M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
    [CrossRef]
  10. V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneties in Markov random process approximation,” Sov. Phys. JETP 29, 1133–1147 (1969).
  11. E. A. Novikov, “The solution of some variational differential equations,” Usp. Mat. Nauk 16, 135–141 (1961).
  12. K. K. Sabelfeld and V. I. Tatarskii, “Approximate calculation of Wiener continual integrals,” Sov. Phys. Dokl. 243, 905–908 (1978).
  13. M. I. Charnotskii, “Coherent channel expansion in the theory of wave propagation in turbulence,” in Proceedings of the Progress in Electromagnetic Research Symposium (PIERS 95), J.A.Kong, ed. (University of Washington, 1995), p. 110.
  14. I. G. Yakushkin, “Intensity fluctuations of wave field scattered at small angles,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 28, 535–565 (1985). (In Russian).
  15. G. Y. Wang and R. Dashen, “Intensity moments for waves in random-media: three-order standard asymptotic calculation,” J. Opt. Soc. Am. A 10, 1226–1232 (1993).
    [CrossRef]
  16. V. U. Zavorotnyy, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
    [CrossRef]

2010 (1)

2009 (1)

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203-1–734203-8 (2009).

2006 (2)

G. J. Baker, “Gaussian beam weak scintillation: Low-order turbulence effects and applicability of the Rytov method,” J. Opt. Soc. Am. A 23, 395–417 (2006).
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

1995 (1)

M. I. Charnotskii, “Coherent channel expansion in the theory of wave propagation in turbulence,” in Proceedings of the Progress in Electromagnetic Research Symposium (PIERS 95), J.A.Kong, ed. (University of Washington, 1995), p. 110.

1994 (1)

M. I. Charnotskii, “Asymptotic analysis of finite beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

1993 (2)

G. Y. Wang and R. Dashen, “Intensity moments for waves in random-media: three-order standard asymptotic calculation,” J. Opt. Soc. Am. A 10, 1226–1232 (1993).
[CrossRef]

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
[CrossRef]

1992 (1)

V. A. Banakh and I. N. Smalikho, “Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 233–237 (1992).

1988 (1)

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4. Wave Propagation through Random Media (Springer, 1988).

1985 (1)

I. G. Yakushkin, “Intensity fluctuations of wave field scattered at small angles,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 28, 535–565 (1985). (In Russian).

1979 (2)

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of 5th Symposium on Laser Propagation in the Atmosphere, (USSR Acad. Sci., Siberia, 1979), pp. 74–78.

V. U. Zavorotnyy, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
[CrossRef]

1978 (1)

K. K. Sabelfeld and V. I. Tatarskii, “Approximate calculation of Wiener continual integrals,” Sov. Phys. Dokl. 243, 905–908 (1978).

1969 (1)

V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneties in Markov random process approximation,” Sov. Phys. JETP 29, 1133–1147 (1969).

1961 (1)

E. A. Novikov, “The solution of some variational differential equations,” Usp. Mat. Nauk 16, 135–141 (1961).

Andrews, L. C.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Baker, G. J.

Banakh, V. A.

V. A. Banakh and I. N. Smalikho, “Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 233–237 (1992).

Charnotskii, M. I.

M. I. Charnotskii, “Beam scintillations for ground-to-space propagation II. Gaussian beam scintillation,” J. Opt. Soc. Am. A 27, 2180–2187 (2010).
[CrossRef]

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203-1–734203-8 (2009).

M. I. Charnotskii, “Coherent channel expansion in the theory of wave propagation in turbulence,” in Proceedings of the Progress in Electromagnetic Research Symposium (PIERS 95), J.A.Kong, ed. (University of Washington, 1995), p. 110.

M. I. Charnotskii, “Asymptotic analysis of finite beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
[CrossRef]

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of 5th Symposium on Laser Propagation in the Atmosphere, (USSR Acad. Sci., Siberia, 1979), pp. 74–78.

Dashen, R.

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
[CrossRef]

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4. Wave Propagation through Random Media (Springer, 1988).

Novikov, E. A.

E. A. Novikov, “The solution of some variational differential equations,” Usp. Mat. Nauk 16, 135–141 (1961).

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4. Wave Propagation through Random Media (Springer, 1988).

Sabelfeld, K. K.

K. K. Sabelfeld and V. I. Tatarskii, “Approximate calculation of Wiener continual integrals,” Sov. Phys. Dokl. 243, 905–908 (1978).

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Smalikho, I. N.

V. A. Banakh and I. N. Smalikho, “Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 233–237 (1992).

Tatarskii, V. I.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
[CrossRef]

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4. Wave Propagation through Random Media (Springer, 1988).

K. K. Sabelfeld and V. I. Tatarskii, “Approximate calculation of Wiener continual integrals,” Sov. Phys. Dokl. 243, 905–908 (1978).

V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneties in Markov random process approximation,” Sov. Phys. JETP 29, 1133–1147 (1969).

Wang, G. Y.

Yakushkin, I. G.

I. G. Yakushkin, “Intensity fluctuations of wave field scattered at small angles,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 28, 535–565 (1985). (In Russian).

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
[CrossRef]

Zavorotnyy, V. U.

V. U. Zavorotnyy, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
[CrossRef]

Atmos. Oceanic Opt. (1)

V. A. Banakh and I. N. Smalikho, “Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere,” Atmos. Oceanic Opt. 5, 233–237 (1992).

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

I. G. Yakushkin, “Intensity fluctuations of wave field scattered at small angles,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 28, 535–565 (1985). (In Russian).

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Radiophys. Quantum Electron. (1)

V. U. Zavorotnyy, “Strong fluctuations of the wave intensity behind a randomly inhomogeneous layer,” Radiophys. Quantum Electron. 22, 352–354 (1979).
[CrossRef]

Sov. Phys. Dokl. (1)

K. K. Sabelfeld and V. I. Tatarskii, “Approximate calculation of Wiener continual integrals,” Sov. Phys. Dokl. 243, 905–908 (1978).

Sov. Phys. JETP (1)

V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneties in Markov random process approximation,” Sov. Phys. JETP 29, 1133–1147 (1969).

Usp. Mat. Nauk (1)

E. A. Novikov, “The solution of some variational differential equations,” Usp. Mat. Nauk 16, 135–141 (1961).

Waves Random Media (1)

M. I. Charnotskii, “Asymptotic analysis of finite beam scintillations in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

Other (5)

M. I. Charnotskii, “Strong intensity fluctuations of finite light beams in a turbulent atmosphere,” in Proceedings of 5th Symposium on Laser Propagation in the Atmosphere, (USSR Acad. Sci., Siberia, 1979), pp. 74–78.

M. I. Charnotskii, J. Gozani, V. I. Tatarskii, and V. U. Zavorotny, “Wave propagation theories in random media based on the path-integral approach,” in Progress in Optics, Vol. XXXII, E.Wolf, ed. (North-Holland, 1993).
[CrossRef]

M. I. Charnotskii, “Laser beam propagation in the low-order turbulence: Exact solution,” Proc. SPIE 7324, 734203-1–734203-8 (2009).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. Vol. 4. Wave Propagation through Random Media (Springer, 1988).

M. I. Charnotskii, “Coherent channel expansion in the theory of wave propagation in turbulence,” in Proceedings of the Progress in Electromagnetic Research Symposium (PIERS 95), J.A.Kong, ed. (University of Washington, 1995), p. 110.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (67)

Equations on this page are rendered with MathJax. Learn more.

2 i k u ( r , z ) + Δ u ( r , z ) + k 2 ε ( r , z ) u ( r , z ) = 0 , u ( r , z = 0 ) = u 0 ( r ) .
u ( r , L ) = d 2 r 0 u 0 ( r 0 ) G ( r 0 , 0 ; r , L ) ,
G ( r 0 , Z 0 ; r , Z ) = D 2 v ( ζ ) exp [ i k 2 Z 0 Z d z v 2 ( z ) + i k ( r r 0 ) 2 2 | Z Z 0 | + i k 2 Z 0 Z d z ε ( r ( z ) , z ) ] δ 2 ( Z 0 Z d z v ( z ) ) ,
r ( z ) = r z Z 0 Z Z 0 + r 0 Z z Z Z 0 + z Z d ζ v ( ζ ) ,
D 2 v ( z ) exp [ i k 2 Z 0 Z d z v 2 ( z ) ] = 1 .
ε ( r + ρ , z ) ε ( r , z ) = A ( ρ , z ) δ ( z z ) .
H ( ρ , z ) = 1 π [ A ( 0 , z ) A ( ρ , z ) ] = 2 d 2 κ Φ ε ( κ , z ) [ 1 cos ( κ ρ ) ] ,
Φ ε ( κ , z ) = 0.033 C ε 2 ( z ) κ 11 3 ,
H ( ρ , z ) = 0.464 C ε 2 ( z ) ρ 5 3 .
z Γ 2 ( R , ρ , z ) i k R ρ Γ 2 ( R , ρ , z ) + π k 2 4 H ( ρ , z ) Γ 2 ( R , ρ , z ) = 0 .
Γ 2 ( R , ρ , z ) = k 2 4 π 2 L 2 d 2 R 0 d 2 ρ 0 Γ 2 ( R 0 , ρ 0 , z = 0 ) × exp [ i k z ( R R 0 ) ( ρ ρ 0 ) π k 2 4 0 z d ζ H ( ρ 0 ( 1 ζ z ) + ρ ζ z , ζ ) ] .
Γ 2 ( R , ρ , z ) = d 2 R 0 d 2 ρ 0 Γ 2 ( R 0 , ρ 0 , z = 0 ) exp [ i k z ( R R 0 ) ( ρ ρ 0 ) ] D 2 v ( ζ ) D 2 v ( ζ ) exp [ i k 2 0 z d ζ ( v ( ζ ) v ( ζ ) ) ] δ ( 0 z d ζ v ( ζ ) ) δ ( 0 z d ζ v ( ζ ) ) exp [ π k 2 4 0 z d ζ H ( ρ 0 ( 1 ζ z ) + ρ ζ z + ζ z d η [ v ( η ) v ( η ) ] , ζ ) ] .
V ( ζ ) = 1 2 [ v 1 ( ζ ) + v 2 ( ζ ) ] , v ( ζ ) = v 1 ( ζ ) v 2 ( ζ ) ,
D 2 V ( ζ ) exp [ i k 0 z d ζ V ( ζ ) v ( ζ ) ] δ ( 0 z d ζ V ( ζ ) ) .
δ ( 0 z d ζ V ( ζ ) ) = 1 4 π 2 d 2 p exp ( i p 0 z d ζ V ( ζ ) ) ,
1 4 π 2 d 2 p D 2 V ( ζ ) exp [ i k 0 z d ζ V ( ζ ) ( v ( ζ ) + p k ) ] .
δ { w ( ζ ) } = D 2 v ( ζ ) exp [ i k 0 z d ζ v ( ζ ) w ( ζ ) ] .
δ N ( A ) = n = 1 N δ 1 ( A n ) = 1 ( 2 π ) N d p 1 . d p N exp ( i n = 1 N A n p n )
D 2 w ( ζ ) F { w ( ζ ) } δ { w ( ζ ) } = F { 0 } .
D 2 V ( ζ ) exp [ i k 0 z d ζ V ( ζ ) v ( ζ ) ] δ ( 0 z d ζ V ( ζ ) ) = 1 4 π 2 d 2 p δ ( v ( ζ ) + p k ) ,
Γ 2 ( R , ρ , z ) = 1 4 π 2 d 2 R 0 d 2 ρ 0 Γ 2 ( R 0 , ρ 0 , z = 0 ) exp [ i k z ( R R 0 ) ( ρ ρ 0 ) ] d 2 p δ ( p z k ) exp [ π k 2 4 0 z d ζ H ( ρ 0 ( 1 ζ z ) + ρ ζ z p z ζ k , ζ ) ] .
I ( R , L ) = k 2 4 π 2 L 2 d 2 R 0 d 2 ρ Γ 2 ( R 0 , ρ ) exp [ i k L ( R 0 R ) ρ π k 2 4 0 L T d z H ( ρ , z ) ] .
r C = ( 0.73 k 2 0 L T d z C ε 2 ( z ) ) 3 5 ,
I ( R , L ) = k 2 4 π 2 L 2 d 2 R 0 d 2 ρ Γ 2 ( R 0 , ρ ) exp [ i k L ( R 0 R ) ρ 1 2 ( ρ r C ) 5 3 ] .
σ I 2 = I 2 ( 0 , L ) I ( 0 , L ) 2 I ( 0 , L ) 2 ,
Γ 4 ( R , r 1 , r 2 , ρ , z ) = u ( R + r 1 + r 2 2 + ρ 4 , z ) u ( R r 1 + r 2 2 + ρ 4 , z ) u * ( R + r 1 r 2 2 ρ 4 , z ) u * ( R r 1 r 2 2 ρ 4 , z ) ,
z Γ 4 ( R , r 1 , r 2 , ρ , z ) i k ( R ρ + r 1 r 2 ) Γ 4 + π k 2 4 Ψ ( r 1 , r 2 , ρ , z ) Γ 4 = 0 ,
Ψ ( r 1 , r 2 , ρ , z ) = H ( r 1 + 1 2 ρ , z ) + H ( r 1 1 2 ρ , z ) + H ( r 2 + 1 2 ρ , z ) + H ( r 2 1 2 ρ , z ) H ( r 1 + r 2 , z ) H ( r 1 r 2 , z ) .
Γ 4 ( R , r 1 , r 2 , ρ , z ) = d 2 R 0 d 2 r 10 d 2 r 20 d 2 ρ 0 Γ 2 ( R 0 , r 10 , r 20 , ρ 0 , 0 ) exp [ i k z ( R R 0 ) ( ρ ρ 0 ) + i k z ( r 1 r 10 ) ( r 2 r 20 ) ] × D 2 v ( ζ ) D 2 v ( ζ ) D 2 w ( ζ ) D 2 w ( ζ ) exp [ i k 2 0 z d ζ [ v 2 ( ζ ) + w 2 ( ζ ) v 2 ( ζ ) w 2 ( ζ ) ] ] × δ ( 0 z d ζ v ( ζ ) ) δ ( 0 z d ζ v ( ζ ) ) δ ( 0 z d ζ w ( ζ ) ) δ ( 0 z d ζ w ( ζ ) ) exp [ π k 2 4 0 z d ζ Ψ ( r ̃ 1 ( ζ ) , r ̃ 2 ( ζ ) , ρ ̃ ( ζ ) , ζ ) ] ,
r ̃ 1 ( ζ ) = r 10 ( 1 ζ z ) + r 1 ζ z + 1 2 ζ z d η [ v ( η ) w ( η ) v ( η ) + w ( η ) ] ,
r ̃ 2 ( ζ ) = r 20 ( 1 ζ z ) + r 2 ζ z + 1 2 ζ z d η [ v ( η ) w ( η ) + v ( η ) w ( η ) ] ,
ρ ̃ ( ζ ) = ρ 0 ( 1 ζ z ) + ρ ζ z + ζ z d η [ v ( η ) + w ( η ) v ( η ) w ( η ) ] .
V ( ζ ) = 1 4 [ v ( ζ ) + w ( ζ ) + v ( ζ ) + w ( ζ ) ] , v ( ζ ) = v ( ζ ) + w ( ζ ) v ( ζ ) w ( ζ ) ,
v 1 ( ζ ) = 1 2 [ v ( ζ ) w ( ζ ) v ( ζ ) + w ( ζ ) ] , v 2 ( ζ ) = 1 2 [ v ( ζ ) w ( ζ ) + v ( ζ ) w ( ζ ) ] ,
Γ 4 ( R , r 1 , r 2 , ρ , z ) = k 2 4 π 2 z 2 d 2 R 0 d 2 r 10 d 2 r 20 d 2 ρ 0 Γ 2 ( R 0 , r 10 , r 20 , ρ 0 , 0 ) exp [ i k z ( R R 0 ) ( ρ ρ 0 ) + i k z ( r 1 r 10 ) ( r 2 r 20 ) ] × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) δ ( 0 z d ζ v 1 ( ζ ) ) δ ( 0 z d ζ v 2 ( ζ ) ) exp [ i k 0 z d ζ v 1 ( ζ ) v 2 ( ζ ) π k 2 4 0 z d ζ Ψ ( r ¯ 1 ( ζ ) , r ¯ 2 ( ζ ) , ρ ¯ ( ζ ) , ζ ) ] ,
ρ ¯ ( ζ ) = ρ 0 ( 1 ζ z ) + ρ ζ z , r ¯ i ( ζ ) = r i 0 ( 1 ζ z ) + r i ζ z + ζ z d η v i ( η ) , i = 1 , 2 .
I 2 ( 0 , L ) = k 2 4 π 2 L 2 d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) ] × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) exp [ π k 2 4 0 L T d z Ψ ( r 1 ( z ) , r 2 ( z ) , ρ , z ) ] ,
r i ( z ) = r i + z L d ζ v i ( ζ ) , i = 1 , 2 .
I ( R , L ) 0 = k 2 4 π 2 L 2 d 2 R 0 d 2 ρ 0 Γ 2 ( R 0 , ρ 0 ) exp [ i k L ( R 0 R ) ρ 0 ]
σ I 2 = k 2 4 π 2 L 2 I 0 2 ( 0 , L ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) ] × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) exp { π k 2 4 0 L T d z [ H ( r 1 ( z ) + ρ 2 , z ) + H ( r 1 ( z ) ρ 2 , z ) ] } { exp [ π k 2 4 0 L T d z Q ( r 1 ( z ) , r 2 ( z ) , ρ ) ] 1 } ,
Q ( r 1 , r 2 , ρ ) = H ( r 2 ( z ) + 1 2 ρ , z ) + H ( r 2 ( z ) 1 2 ρ , z ) H ( r 1 ( z ) + r 2 ( z ) , z ) H ( r 1 ( z ) r 2 ( z ) , z ) .
σ I 2 = W 1 + W 2 + ,
W 1 = π k 2 4 k 2 4 π 2 L 2 I 0 2 ( 0 , L ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) ] × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) 0 L T d z [ H ( r 1 ( z ) + r 2 ( z ) , z ) + H ( r 1 ( z ) r 2 ( z ) , z ) H ( r 2 ( z ) + 1 2 ρ , z ) H ( r 2 ( z ) 1 2 ρ , z ) ] ,
W 1 = π k 2 k 2 4 π 2 L 2 I 0 2 ( 0 , L ) 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) + i κ r 2 + i κ z L d ζ v 2 ( ζ ) + i k L ( R ρ + r 1 r 2 ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) [ cos ( 1 2 κ ρ ) cos ( κ r 1 + κ z L d ζ v 1 ( ζ ) ) ] .
W 11 = π k 2 ( k 2 4 π 2 L 2 I 0 ( 0 , L ) ) 2 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) + i κ r 2 ] cos ( 1 2 κ ρ ) .
W 12 = π k 2 k 2 4 π 2 L 2 I 0 2 ( 0 , L ) 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) + i κ r 2 + i κ z L d ζ v 2 ( ζ ) + i k L ( R ρ + r 1 r 2 ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) cos ( κ r 1 + κ z L d ζ v 1 ( ζ ) ) ,
D 2 v 2 ( ζ ) δ ( 0 L d z v 2 ( z ) ) exp [ i k 0 L d ζ v 2 ( ζ ) ( v 1 ( ζ ) + i k κ ϑ ( ζ z ) ) ] ,
D 2 v 2 ( ζ ) δ ( 0 L d z v 2 ( z ) ) exp [ i k 0 L d ζ v 2 ( ζ ) ( v 1 ( ζ ) + κ k ϑ ( ζ z ) ) ] = 1 4 π 2 d 2 p δ ( v 1 ( ζ ) + κ k ϑ ( ζ z ) + p k ) .
W 12 = π k 2 ( k 2 4 π 2 L 2 I 0 ( 0 , L ) ) 2 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) + i κ r 2 ] cos ( κ r 1 κ 2 z ( L z ) k L ) .
W 1 = π k 2 ( k 2 4 π 2 L 2 I 0 ( 0 , L ) ) 2 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) + i κ r 2 ] [ cos ( 1 2 κ ρ ) cos ( κ r 1 κ 2 z k ) ] .
Ψ ( r 1 , 0 , 0 , ζ ) = Ψ ( 0 , r 2 , 0 , ζ ) = 0 .
{ r 1 ( ζ ) r C ρ ( ζ ) r C } { r 2 ( ζ ) r C ρ ( ζ ) r C }
exp [ π k 2 4 0 L T d ζ Ψ ( r 1 ( ζ ) , r 2 ( ζ ) , ρ ) ] = exp { π k 2 4 0 L T d ζ [ H ( r 1 ( ζ ) + 1 2 ρ ) + H ( r 1 ( ζ ) 1 2 ρ ) ] } [ 1 π k 2 4 0 L T d ζ Q ( r 1 , r 2 , ρ ) + 1 2 ( π k 2 4 0 L T d ζ Q ( r 1 , r 2 , ρ ) ) 2 + ] ,
exp [ π k 2 4 0 L T d ζ Ψ ( r 1 , r 2 , ρ ) ] = exp { π k 2 4 0 L T d ζ [ H ( r 1 + ρ 2 ) + H ( r 1 ρ 2 ) ] } [ 1 π k 2 4 0 θ L d ζ Q ( r 1 , r 2 , ρ ) + ] + exp { π k 2 4 0 L T d ζ [ H ( r 2 + ρ 2 ) + H ( r 2 ρ 2 ) ] } [ 1 π k 2 4 0 θ L d ζ Q ( r 2 , r 1 , ρ ) + ] exp { π k 2 4 0 L T d ζ [ H ( r 1 + ρ 2 ) + H ( r 1 ρ 2 ) + H ( r 2 + ρ 2 ) + H ( r 2 ρ 2 ) ] } × [ 1 + π k 2 4 0 L T d ζ [ H ( r 1 + r 2 ) + H ( r 1 + r 2 ) ] + ] .
I 2 ( 0 , L ) 1 = M 0 + M 1 + M 2 + .
M 0 = k 2 4 π 2 L 2 d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) ] × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) exp { π k 2 4 0 L T d z [ H ( r 1 + ρ 2 + z L d ζ v 1 ( ζ ) , z ) + H ( r 1 ρ 2 + z L d ζ v 1 ( ζ ) , z ) ] } .
M 0 = ( k 2 4 π 2 L 2 ) 2 d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) ] exp { π k 2 4 0 L T d z [ H ( r 1 + ρ 2 , z ) + H ( r 1 ρ 2 , z ) ] } .
Γ 4 ( R , r 1 , r 2 , ρ , 0 ) = Γ 2 ( R + r 2 2 , r 1 + ρ 2 , 0 ) Γ 2 ( R r 2 2 , r 1 + ρ 2 , 0 ) ,
M 0 = I ( 0 , L ) 2 .
σ I 2 = 1 + 2 M 1 I ( 0 , L ) 2 + 2 M 2 I ( 0 , L ) 2 + .
M 1 = π k 2 4 k 2 4 π 2 L 2 0 L T d z d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp [ i k L ( R ρ + r 1 r 2 ) ] × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 1 ( ζ ) v 2 ( ζ ) ] δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) exp { π k 2 4 0 L T d ζ [ H ( r 1 + ρ 2 + ζ L d v v 1 ( η ) , ζ ) + H ( r 1 ρ 2 + ζ L d η v 1 ( η ) , ζ ) ] } [ H ( r 1 ( z ) + r 2 ( z ) , z ) + H ( r 1 ( z ) r 2 ( z ) , z ) H ( r 2 ( z ) + ρ 2 , z ) H ( r 2 ( z ) ρ 2 , z ) ] ,
M 1 = π k 2 k 2 4 π 2 L 2 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) × D 2 v 1 ( ζ ) D 2 v 2 ( ζ ) exp [ i k 0 L d ζ v 2 ( ζ ) ( v 1 ( ζ ) + κ k ϑ ( L z ) ) + i k L ( R ρ + r 1 r 2 ) + i κ r 2 ] exp { π k 2 4 0 L T d ζ [ H ( r 1 + ρ 2 + ζ L d v v 1 ( η ) , ζ ) + H ( r 1 ρ 2 + ζ L d η v 1 ( η ) , ζ ) ] } δ ( 0 L d ζ v 1 ( ζ ) ) δ ( 0 L d ζ v 2 ( ζ ) ) [ cos ( 1 2 κ ρ ) cos ( κ r 1 + κ z L d ζ v 1 ( ζ ) ) ] .
M 1 = π k 2 ( k 2 4 π 2 L 2 ) 2 0 L T d z d 2 κ Φ ε ( κ , z ) d 2 R d 2 r 1 d 2 r 2 d 2 ρ Γ 4 ( R , r 1 , r 2 , ρ , 0 ) exp { π k 2 4 0 L T d ζ [ H ( r 1 + ρ 2 κ k min ( z , ζ ) , ζ ) + H ( r 1 ρ 2 κ k min ( z , ζ ) , ζ ) ] } exp [ i k L ( R ρ + r 1 r 2 ) + i κ r 2 ] [ cos ( κ ρ 2 ) cos ( κ r 1 κ 2 z k ) ] .
G ( r 0 , Z 0 ; r , Z ) = d 2 r 1 d 2 r N 1 n = 1 N G ( r n 1 , Z 0 + ( n 1 ) Δ z ; r n , Z 0 + n Δ z ) ,
G ( r , z , r , z + Δ z ) = k 2 π i Δ z exp [ i k ( r r ) 2 2 Δ z + i k 2 ε ( r , z ) Δ z + O ( Δ z 2 ) ] .
G ( r , Z ; r 0 , Z 0 ) = ( k 2 π i Δ z ) N d 2 r 1 d 2 r N 1 exp [ i k 2 n = 1 N ( r n r n 1 ) 2 Δ z + i k 2 n = 1 N ε ( r n 1 , Z 0 + n Δ z ) ] .
G ( r , Z ; r 0 , Z 0 ) = D 2 r ( z ) exp [ i k 2 Z 0 Z d z ( d r d z ) 2 + i k 2 Z 0 Z d z ε ( r ( z ) , z ) ] δ 2 [ r ( Z 0 ) r 0 ] δ 2 [ r ( Z ) r ] .

Metrics